Hollow cylindrical silicon metamaterial sensor and its parametric inversion model

Abstract

A hollow cylindrical silicon metamaterial sensor and its parameter inversion model are designed. The sensor breaks the symmetry by adjusting the asymmetric parameters, converting the symmetrically protected continuous domain bound state into a high-quality factor quasi-continuous domain bound state mode. A parameter inversion model is designed for the hollow cylindrical metamaterial sensor. The ResNet50-1D and LSTM parallel feature extraction architecture are combined with the fusion module to realize efficient extraction of multi-scale spectral features. A differentiable physical information neural network head is introduced to establish peak consistency constraints and ensure the physical credibility of the inversion structure parameters, which is particularly suitable for efficient development of high-sensitivity terahertz sensing chips.

Description

Technical Field

[0001] This application relates to the fields of optical sensing metasurfaces and deep learning, and in particular to hollow cylindrical metamaterial sensors and their parameter inversion models. Background Technology

[0002] In recent years, driven by significant advancements in biosensor technology, terahertz (THz) sensing has once again attracted attention. Terahertz (THz) spectroscopy, due to its non-ionizing, label-free, tissue-penetrating, and highly sensitive properties to molecular configuration and dielectric environment, has become a key technology for the detection of biomolecules, viruses, and cells. All-dielectric metamaterial sensors based on continuous-domain bound states (BIC) can be transformed into quasi-BIC (QBIC) modes through symmetry breaking, achieving ultra-high Q values, extremely narrow resonance peaks, and strong field localization. They exhibit ultra-high sensitivity to changes in the refractive index of the microenvironment, making them a preferred structure for constructing highly sensitive terahertz biosensor chips.

[0003] Furthermore, with the continuous advancement of terahertz technology, the design of terahertz metamaterials faces increasingly complex challenges. The design of traditional metamaterial sensors relies on numerical simulation, empirical trial and error, and iterative optimization, which suffers from long cycles, high costs, and difficulty in quickly matching the target optical response. In particular, for hollow cylindrical QBIC structures, the resonance peak position, linewidth, Q value, and geometric variables such as lattice constant, inner and outer radii, structural height, and asymmetry parameters exhibit strong nonlinear correlations, making it difficult to quickly deduce the optimal structural parameters through forward design.

[0004] Deep learning technologies, especially deep neural networks (DNNs), convolutional neural networks (CNNs), and generative adversarial networks (dGANs), have become important tools for solving this problem. They can automatically learn complex nonlinear relationships from large amounts of experimental or simulation data. Deep learning not only accelerates the design optimization process but also effectively predicts electromagnetic responses and performs inverse design in metamaterial design, avoiding the complex physical analysis and time-consuming optimization processes of traditional methods. However, current deep learning models still have significant shortcomings in the parameter inversion of hollow cylindrical QBIC sensors. Feature extraction mismatch: QBIC resonance is an extremely narrow and sharp Fano peak. Ordinary deep learning models have difficulty capturing both the extremely sharp Fano resonance peak (local feature) in the transmission spectrum and the wide range of contours of the entire spectrum (global temporal dependence). Lack of physical consistency: Purely data-driven models are like a "black box". Their prediction results often only approximate the training data numerically, but lack clear physical mechanism constraints. This may result in the predicted structural parameters failing to produce the physical characteristics that the target spectrum should have (such as resonance peak position and linewidth), i.e., "physically unreasonable". Poor generalization and robustness: It is sensitive to simulation noise and small sample data, prone to overfitting, and has weak extrapolation ability in the parameter space of hollow cylindrical structures.

[0005] Currently, developing a new method for efficiently, accurately, and physically reliably inverting the parameters of hollow cylindrical quasi-BIC terahertz metamaterial sensors has become a key technical challenge that urgently needs to be addressed in this field. Summary of the Invention

[0006] This application provides a hollow cylindrical metamaterial sensor and its parameter inversion model. It proposes a hollow cylindrical metamaterial sensor and a metamaterial sensor parameter inversion method that integrates deep learning and physical consistency constraints. This method can efficiently and collaboratively capture the unique, extremely narrow resonance peaks of the quasi-BIC and the temporal evolution of the entire spectrum. At the same time, it ensures that the predicted structural parameters can not only numerically fit the label, but also strictly guarantee that the generated spectrum is consistent with the input target spectrum in key physical features (such as peak position and peak shape).

[0007] In a first aspect, embodiments of this application provide a hollow cylindrical metamaterial sensor, comprising: at least one metamaterial sensing unit, wherein each metamaterial sensing unit includes a silicon dioxide substrate and at least one hollow cylindrical silicon structure located on the silicon dioxide substrate; Each hollow cylindrical silicon structure is a hollow cylindrical ring composed of a first semicircular ring and a second semicircular ring arranged at intervals. The second semicircular ring is further divided by gaps into symmetrically distributed first and second arc-shaped rings. The central angle of the first semicircular ring is 180°, and the central angles of the first and second arc-shaped rings are both 90°. The distance between the first and second arc-shaped rings is defined as an asymmetric parameter. Secondly, embodiments of this application provide a method for constructing a parameter inversion model for a hollow cylindrical metamaterial sensor, and the design of the hollow cylindrical metamaterial sensor includes the following steps: S1: Obtain the training dataset, which includes at least one set of transmission curves of hollow cylindrical metamaterial sensors and corresponding hollow cylindrical metamaterial sensor variables for each transmission curve. The hollow cylindrical metamaterial sensor variables include asymmetric parameters, the height of the hollow cylindrical silicon structure, the outer ring radius of the hollow cylindrical silicon structure, and the inner ring radius of the hollow cylindrical silicon structure. S2: The training dataset is input into the parameter inversion framework for training until the loss function meets the set conditions to obtain the parameter inversion model. The parameter inversion framework includes an input layer, a ResNet50 branch, an LSTM branch, a feature fusion module, a linear layer, a PINN physical head, and an output layer. The input layer normalizes the training dataset to obtain one-dimensional sequence data. The one-dimensional sequence data is input into the ResNet50 branch and the LSTM branch to obtain ResNet50 extracted features and LSTM extracted features, respectively. The ResNet50 extracted features and LSTM extracted features are input into the fusion module to output fused features. The fused features are input into the linear layer to obtain the prediction results. The training dataset is input into the PINN physical head to construct the target peak map, and the prediction results are input into the PINN physical head to construct the prediction peak map. The loss function of the parametric inversion framework is the supervised regression loss and the peak loss. Figure 1 The weighting function for consistency loss, and the supervised regression loss are constructed based on the prediction results and the training dataset, with peak values... Figure 1 The consistency loss is constructed based on the target peak map and the predicted peak map.

[0008] Thirdly, embodiments of this application provide a parameter design method for a hollow cylindrical metamaterial sensor, comprising the following steps: The transmission curve of the hollow cylindrical metamaterial sensor is input into the parameter inversion model to obtain the prediction results, which include the asymmetric parameters of the hollow cylindrical metamaterial sensor, the height of the hollow cylindrical silicon structure, the outer ring radius of the hollow cylindrical silicon structure, and the inner ring radius of the hollow cylindrical silicon structure.

[0009] The main contributions and innovations of this invention are as follows: This application presents a hollow cylindrical metamaterial sensor composed of a 180° semi-circular annulus and two sets of 90° circular arc cylinders. By precisely controlling the in-plane symmetry breaking through the asymmetric parameter 'a', the symmetry-protected BIC is efficiently converted into an electromagnetic hybrid QBIC mode, achieving an ultra-high quality factor resonance with a Q value of ≈3397 at 0.917 THz. Simultaneously, it achieves excellent sensing performance with a sensitivity of 100 GHz / RIU and a FOM of 361. The resonant frequency band precisely matches the characteristic absorption peak of hepatitis B virus, allowing direct application for highly sensitive, label-free biomolecule detection, thus solving the problems of low Q value and insufficient sensitivity in existing terahertz sensors.

[0010] Meanwhile, this solution designs a parameter inversion model specifically for this hollow cylindrical metamaterial sensor. Addressing the difficulty in capturing the extremely narrow and sharp Fano resonance peak of QBIC, it employs a ResNet50-1D branch to accurately extract local sharp peak features in the spectrum, and an LSTM branch to model long-term temporal dependencies across the entire spectrum. Combined with a multi-head self-attention mechanism, this overcomes the technical limitation of a single network being unable to simultaneously handle both local fine features and global temporal dependencies, significantly improving the efficiency and representational capability of transmission spectrum feature extraction. Furthermore, a dual-branch attention mechanism (channel and spatial) is designed to intelligently allocate weights and deeply fuse convolutional and temporal features, preserving high-value information and suppressing redundant noise. This ensures that the fused features are rich in both local details and global context, greatly improving the network's learning accuracy and stability of the structure-spectral nonlinear mapping relationship. Furthermore, the parameter inversion model of this scheme also constructs a differentiable physical mapping from structural parameters to resonance peak characteristics. Closed-loop constraints are achieved through consistency verification between the target peak diagram and the predicted peak diagram, so that the structural parameters output by inversion not only closely approximate the label numerically, but also meet key physical laws such as peak position, linewidth, and Q value. This avoids physically unreasonable prediction results from the root, and improves the reliability and interpretability of the inversion.

[0011] Details of one or more embodiments of this application are set forth in the following drawings and description to make other features, objects and advantages of this application more readily apparent. Attached Figure Description

[0012] The accompanying drawings, which are included to provide a further understanding of this application and form part of this application, illustrate exemplary embodiments and are used to explain this application, but do not constitute an undue limitation of this application. In the drawings: Figure 1 This is a schematic diagram of the structure of a hollow cylindrical metamaterial sensor according to an embodiment of this application.

[0013] Figure 2 This is a schematic diagram of the framework of a parameter inversion model according to an embodiment of this application.

[0014] Figure 3 This is a schematic diagram of the framework of a fusion module for a parameter inversion model according to an embodiment of this application.

[0015] Figure 4 It is the transmission spectrum of the metasurface of the sensor under symmetric and asymmetric conditions.

[0016] Figure 5 This is a radiation power diagram of different multipoles in Cartesian coordinates (a = 7 μm).

[0017] Figure 6 This is a diagram of the electromagnetic field distribution at 0.917 THz.

[0018] Figure 7 These are the transmission curves of sensors with different asymmetric parameters.

[0019] Figure 8 These are THz transmission spectra with different asymmetric parameters.

[0020] Figure 9 These are the transmission curves of sensors with different outer ring radii of hollow cylindrical silicon structures.

[0021] Figure 10 These are the transmission curves of sensors with different heights of hollow cylindrical silicon structures.

[0022] Figure 11 These are simulated transmission spectra of sensors at different refractive indices.

[0023] Figure 12 It is a graph showing the relationship between different refractive indices and spectral frequencies. Detailed Implementation

[0024] Exemplary embodiments will now be described in detail, examples of which are illustrated in the accompanying drawings. When the following description relates to the drawings, unless otherwise indicated, the same numerals in different drawings denote the same or similar elements. The embodiments described in the following exemplary embodiments do not represent all embodiments consistent with one or more embodiments of this specification. Rather, they are merely examples of apparatuses and methods consistent with some aspects of one or more embodiments of this specification as detailed in the appended claims.

[0025] It should be noted that the steps of the corresponding methods are not necessarily performed in the order shown and described in this specification in other embodiments. In some other embodiments, the methods may include more or fewer steps than described in this specification. Furthermore, a single step described in this specification may be broken down into multiple steps in other embodiments; and multiple steps described in this specification may be combined into a single step in other embodiments.

[0026] Example 1 like Figure 1 As shown, this solution provides a hollow cylindrical metamaterial sensor, which includes: At least one metamaterial sensing unit, wherein each metamaterial sensing unit includes a silicon dioxide substrate and at least one hollow cylindrical silicon structure located on the silicon dioxide substrate; Each hollow cylindrical silicon structure is a hollow cylindrical ring composed of a first semicircular ring and a second semicircular ring arranged at intervals. The second semicircular ring is further divided by gaps into symmetrically distributed first and second arc-shaped rings. The central angle of the first semicircular ring is 180°, and the central angles of the first and second arc-shaped rings are both 90°. The distance between the first and second arc-shaped rings is defined as an asymmetric parameter.

[0027] The hollow cylindrical metamaterial sensor provided in this solution breaks the original central rotational symmetry and mirror symmetry of hollow cylindrical silicon through asymmetric parameters, achieving symmetry breaking and degenerating the intrinsic symmetry-protected BIC mode into an excitable, narrow-linewidth, high-Q quasi-BIC (QBIC) resonance. In other words, this solution breaks the intrinsic geometric symmetry of the hollow cylindrical structure by adjusting the magnitude of the asymmetric parameters, realizing the transformation from continuous-domain bound states (BIC) to quasi-BIC (QBIC) resonance modes, thereby controlling the resonance spectral characteristics and sensing performance of the sensor.

[0028] In some embodiments, the asymmetric parameter a ranges from 0 to 28 µm.

[0029] Preferably, the asymmetric parameter a is 7 µm.

[0030] In some embodiments, the first and second semicircular ring cylinders have the same outer and inner diameters, and the central angle of the second semicircular ring cylinder is also 180°. That is, this solution involves cutting along the first central axis to form the first and second semicircular ring cylinders within a complete hollow ring, and then cutting along the second central axis on the second semicircular ring cylinder to form the first and second arc-shaped cylinders, wherein the first and second central axes are perpendicularly arranged.

[0031] In some embodiments, as the outer diameter of the outer ring radius of the hollow cylindrical silicon structure increases, the resonance peak of the hollow cylindrical metamaterial sensor undergoes a redshift, and the linewidth of the resonance peak narrows.

[0032] In some embodiments, the outer ring radius of the hollow cylindrical silicon structure is 60~80μm, selected as 60μm, 65μm, 70μm, 75μm and 80μm.

[0033] In some embodiments, the inner ring radius of the hollow cylindrical silicon structure is 30–50 µm, preferably 42 µm.

[0034] In some embodiments, as the height of the hollow cylindrical silicon structure increases, the resonance peak of the hollow cylindrical metamaterial sensor undergoes a redshift.

[0035] In some embodiments, the height of the hollow cylindrical silicon structure is 34~50μm, selected as 34μm, 38μm, 42μm, 46μm and 50μm.

[0036] Therefore, this scheme can precisely locate the QBIC resonant peak to the target operating band by adjusting the outer diameter of the first semi-circular ring and the height of the hollow cylindrical silicon structure, which provides a solid theoretical and experimental basis for customizing the operating frequency of sensors for specific applications.

[0037] In some embodiments, the relative permittivity of the silicon dioxide substrate is 3.9.

[0038] In some embodiments, the hollow cylindrical silicon structure is made of silicon with a relative permittivity of 11.9.

[0039] In some embodiments, the distance between the first semicircular ring and the second semicircular ring is defined as the vertical distance, wherein the vertical distance is 20 to 30 µm.

[0040] Preferably, the vertical spacing is 27µm.

[0041] In some embodiments, each metamaterial sensing unit is a square structure with a side length of (200–220 µm). Preferably, the side length is 210 µm.

[0042] In some embodiments, the height of the silicon dioxide substrate is (80–90 µm). Preferably, the height of the silicon dioxide substrate is 84 µm.

[0043] Example 2 like Figure 2 As shown, this solution provides a method for constructing a parameter inversion model for hollow cylindrical metamaterial sensors. This parameter inversion model for hollow cylindrical metamaterial sensors integrates deep learning and physical consistency constraints, aiming to enable the network to simultaneously and accurately capture the unique, extremely narrow resonance peaks (local key modes) and the temporal evolution of the entire spectrum (long-range dependence) of the quasi-BIC, and to break the "black box" limitation of pure data-driven models. This allows the network to not only numerically fit the labels of the structural parameters predicted by the network, but also to strictly ensure that the generated spectrum is consistent with the input target spectrum in key physical features (such as peak position and peak shape).

[0044] Specifically, the method for constructing the parameter inversion model for hollow cylindrical metamaterial sensors proposed in this scheme includes the following steps: A hollow cylindrical metamaterial sensor is applicable, comprising at least one metamaterial sensing unit, wherein each metamaterial sensing unit comprises a silicon dioxide substrate and at least one hollow cylindrical silicon structure located on the silicon dioxide substrate; each hollow cylindrical silicon structure is a hollow cylindrical ring composed of a first semi-circular ring column and a second semi-circular ring column arranged at intervals, and the second semi-circular ring column is further divided by a gap into symmetrically distributed first arc column and second arc column, the central angle of the first semi-circular ring column is 180°, the central angle of the first arc column and the second arc column are both 90°, and the distance between the first arc column and the second arc column is defined as an asymmetric parameter; S1: Obtain the training dataset, which includes at least one set of transmission curves of hollow cylindrical metamaterial sensors and corresponding hollow cylindrical metamaterial sensor variables for each transmission curve. The hollow cylindrical metamaterial sensor variables include asymmetric parameters, the height of the hollow cylindrical silicon structure, the outer ring radius of the hollow cylindrical silicon structure, and the inner ring radius of the hollow cylindrical silicon structure. S2: The training dataset is input into the parameter inversion framework for training until the loss function meets the set conditions to obtain the parameter inversion model. The parameter inversion framework includes an input layer, a ResNet50 branch, an LSTM branch, a feature fusion module, a linear layer, a PINN physical head, and an output layer. The input layer normalizes the training dataset to obtain one-dimensional sequence data. The one-dimensional sequence data is input into the ResNet50 branch and the LSTM branch to obtain ResNet50 extracted features and LSTM extracted features, respectively. The ResNet50 extracted features and LSTM extracted features are input into the fusion module to output fused features. The fused features are input into the linear layer to obtain the prediction results. The training dataset is input into the PINN physical head to construct the target peak map, and the prediction results are input into the PINN physical head to construct the prediction peak map. The loss function of the parametric inversion framework is the supervised regression loss and the peak loss. Figure 1 The weighting function for consistency loss, and the supervised regression loss are constructed based on the prediction results and the training dataset, with peak values... Figure 1 The consistency loss is constructed based on the target peak map and the predicted peak map.

[0045] The parameter inversion model proposed in this scheme for hollow cylindrical metamaterial sensors utilizes a parallel structure of two branches: ResNet50 and LSTM. The deep convolutional structure of the ResNet50 branch is used to accurately capture sharp local features such as quasi-BIC resonance peaks in the transmission spectrum, while the LSTM branch is used to model the long-term temporal dependencies of the entire spectral sequence. This solves the inherent contradiction that the single network structure of traditional design models cannot simultaneously achieve both "local precision" and "global temporal accuracy".

[0046] Furthermore, this scheme uses the frequency domain solver of CST electromagnetic simulation software to model and simulate the hollow cylindrical metamaterial sensor to construct a training dataset. Specifically, this scheme uses the frequency domain solver of CST electromagnetic simulation software to collect transmission data of the hollow cylindrical metamaterial sensor and construct transmission curves. The training dataset is constructed using variables of the hollow cylindrical metamaterial sensor, including asymmetric parameters, the height of the hollow cylindrical silicon structure, the outer ring radius of the hollow cylindrical silicon structure, and the inner ring radius of the hollow cylindrical silicon structure.

[0047] In some embodiments, the training dataset includes 1001 sampling points uniformly sampled on the transmission curves of at least one set of hollow cylindrical metamaterial sensors, and hollow cylindrical metamaterial sensor variables corresponding to each transmission curve.

[0048] Furthermore, this scheme collected 900 sets of transmission curves from hollow cylindrical metamaterial sensors and the corresponding variables for each transmission curve. 80% of the data was used as the training set, and the remaining 20% ​​as the test set, resulting in a training set containing 720 sets of data. Additionally, after data collection, the data was prepared into CSV files for training.

[0049] After obtaining the training dataset, this scheme uses the collected training dataset to train a parameter inversion framework. The parameter inversion framework of this scheme is as follows: First, the input layer in this parameter inversion framework normalizes the input data to obtain one-dimensional sequence data. The one-dimensional sequence data is then input into the subsequent ResNet50 branch and LSTM branch for processing to eliminate differences in data dimensions and convert the full-band discrete spectral data into standardized one-dimensional time-series sequence data.

[0050] Secondly, the one-dimensional sequence data is input into the ResNet50 branch and the LSTM branch respectively. The structure of the ResNet50 branch and the LSTM branch is as usual, so it will not be explained again.

[0051] One-dimensional sequence data is input into the ResNet50 branch. The ResNet50 branch captures local key features through convolution operations to obtain the ResNet50 extracted features. However, due to the characteristic of BIC transmission spectrum, which has extremely narrow transmission peaks, there are very few sampling points at the position of the transmission peak. The LSTM network can solve this technical problem by memorizing historical information to capture long-term dependencies in one-dimensional sequence data. The LSTM branch is used to process the temporal information of one-dimensional sequence data.

[0052] Furthermore, the features extracted by ResNet50 and LSTM are input into the fusion module, which outputs fused features. It should be noted that this scheme specifically designs a fusion module that adaptively calculates the importance weights of the ResNet50 and LSTM features through a dual-branch attention mechanism (channel and spatial), thereby achieving intelligent fusion and ensuring that the fused features are rich in both local details and global contextual information.

[0053] The structure of the fusion module in this solution is as follows: Figure 3 As shown: This fusion module includes parallel channel and spatial branches. It innovatively integrates channel attention and spatial attention mechanisms to precisely focus on key information in the feature maps. Simultaneously, by processing dual-temporal features, the fusion module effectively introduces temporal contextual relationships.

[0054] Specifically, in the channel branch processing flow, the input ResNet50 extracted features and LSTM extracted features are first compressed along the channel dimension through a global pooling operation to achieve efficient aggregation of channel information and obtain aggregated spatial features. The specific process is as follows:

[0055] in, This represents the aggregated channel features. and These represent the input features from two different time series: features extracted by ResNet50 and features extracted by LSTM. and Global average pooling and global max pooling are performed sequentially along the channel dimension.

[0056] The channel features were then aggregated. The input is fed into two one-dimensional convolutional layers to generate ResNet50 channel weights and LSTM channel weights corresponding to the features extracted by ResNet50 and LSTM, respectively. The specific calculation expressions for these two weights are as follows:

[0057] in, and These represent the channel weights of ResNet50 and the channel weights of LSTM, respectively. and This corresponds to two independent one-dimensional convolution operations.

[0058] To encourage the weight distribution to focus on more discriminative features, a Softmax function was subsequently introduced. and Normalization is performed to make the sum equal to 1. This process, by comparing and amplifying the relative differences between weights, can accurately highlight the key information of the dual temporal features in the channel dimension. The specific calculation formula of the Softmax function is as follows:

[0059] That is, in the final feature fusion stage, and This represents the normalized ResNet50 channel weights and LSTM channel weights.

[0060] Similarly, in the spatial branch, the normalized ResNet50 spatial weights are derived using the same method. and LSTM spatial weights This method is used to identify key feature regions in the spatial dimension. The ResNet50 temporal weights are obtained by element-wise multiplication of the normalized ResNet50 channel weights and the normalized ResNet50 spatial weights. Similarly, the LSTM temporal weights are obtained by element-wise multiplication of the normalized LSTM spatial weights and the normalized LSTM channel weights. Both the LSTM temporal weights and the ResNet50 temporal weights contain attention information from both the channel and spatial dimensions, thus enabling precise localization of highly discriminative key components within the dual temporal features.

[0061] Finally, by weighting and fusing the LSTM temporal weights and ResNet50 temporal weights with the corresponding features, effective feature integration is achieved, and the output fused feature can be expressed as: ; Where output is the fused feature. and These represent the input features from two different time series: features extracted by ResNet50 and features extracted by LSTM. and This represents the normalized ResNet50 channel weights and LSTM channel weights. These are the normalized ResNet50 spatial weights. The normalized LSTM spatial weights are used to fuse features into a weighted sum of dual temporal features. By constraining the sum of weights to 1, the model automatically focuses on key information and weakens secondary components, thereby achieving efficient feature fusion.

[0062] That is, the channel branch includes global average pooling and global max pooling in the channel dimension, two one-dimensional convolutional layers, and a Softmax function. The features extracted by ResNet50 and LSTM are input into the channel branch and then concatenated after global average pooling and global max pooling in the channel dimension to obtain aggregated channel features. The aggregated channel features are input into two one-dimensional convolutional layers to generate ResNet50 channel weights and LSTM channel weights corresponding to the features extracted by ResNet50 and LSTM. The ResNet50 channel weights and LSTM channel weights are normalized using the Softmax function to obtain normalized ResNet50 channel weights and LSTM channel weights. The spatial branch includes global average pooling and global max pooling in the spatial dimension, two one-dimensional convolutional layers, and a Softmax function. The features extracted by ResNet50 and LSTM are input into the spatial branch and then concatenated after global average pooling and global max pooling in the spatial dimension to obtain aggregated spatial features. The aggregated spatial features are input into two two-dimensional convolutional layers to generate ResNet50 spatial weights and LSTM spatial weights corresponding to the features extracted by ResNet50 and LSTM. The ResNet50 spatial weights and LSTM spatial weights are normalized using the Softmax function to obtain normalized ResNet50 spatial weights and LSTM spatial weights. The ResNet50 temporal weights are obtained by element-wise multiplication of the normalized ResNet50 channel weights and the normalized ResNet50 spatial weights. The LSTM temporal weights are obtained by element-wise multiplication of the normalized LSTM spatial weights and the normalized LSTM channel weights. The fused features are obtained by weighted fusion of the ResNet50 temporal weights and the LSTM temporal weights.

[0063] In this scheme, the channel branch in the fusion module is designed to enhance the focus on channel information, and the spatial branch is designed to enhance the focus on spatial information. This scheme performs spatial dimension pooling and channel dimension pooling on features extracted by ResNet50 and LSTM simultaneously. Then, the spatial dimension pooling and channel dimension pooling are concatenated, followed by weight determination to retain useful information, and finally separation. By adjusting the weights, the more valuable parts of the dual-temporal features are retained, while unimportant or misleading information is suppressed, thereby improving the accuracy and robustness of change detection.

[0064] In some embodiments, spatial dimension pooling includes spatial dimension average pooling and spatial dimension max pooling connected in sequence, and channel dimension pooling includes channel dimension average pooling and channel dimension max pooling connected in sequence.

[0065] Additionally, regarding the PINN physical header in this solution: To ensure that the structural parameters output by the parametric inversion model not only numerically approximate the labeled parameters but also physically match the resonance peak characteristics of the input transmission spectrum, this scheme introduces the PINN physical head and the corresponding physical consistency loss to impose physical prior constraints on the output of the parametric inversion model, thereby improving the interpretability, physical rationality, and generalization robustness of the prediction results.

[0066] The PINN physical head constructs a direct, differentiable mapping from structural parameters to formant characteristics. It processes the prediction results to obtain the number and center frequency (peak position) of the formants. Simultaneously, it extracts differentiable features from the target spectrum of the input training dataset, comparing the two through a loss function to achieve closed-loop verification of physical consistency. This method anchors to the physical essence that determines sensor performance, deeply embedding physical knowledge into the learning process, ensuring the physical rationality of the results and excellent generalization ability with extremely high efficiency.

[0067] In some embodiments, the transmission spectrum from the training dataset is input into the PINN physical head to construct the target peak map using a differentiable soft peaking operator.

[0068] Specifically, the target peak map is obtained by superimposing Gaussian kernels on multiple local maxima in the transmission spectrum of the training dataset, and the calculation formula is as follows: ; in For the target peak diagram, The weight of the k-th peak is calculated based on local peak intensity and sharpness. For the first k The center frequency of a local resonance peak The standard deviation of the Gaussian kernel controls the peak width.

[0069] In some embodiments, the prediction results are input into the lightweight differentiable peak parameter regressor in the PINN physical head to obtain the center frequency and weight of the predicted peak. Then, a predicted peak diagram is constructed based on the center frequency and weight of the predicted peak. The calculation formula is as follows: ; in To predict the peak plot, As weight, To predict the center frequency of the peak, denoted as the standard deviation of the Gaussian kernel.

[0070] Regarding the loss function of this scheme: The loss function of this scheme is the supervised regression loss and the peak loss. Figure 1The weighting function for consistency loss is calculated using the following formula: ; ; ; in To predict the peak plot, For the target peak diagram, For the predicted results, For the true values ​​of the training dataset, For weighted weights, For Peak Figure 1 Sexual damage, To monitor the regression loss, This is the loss function.

[0071] It should be noted that the peak in the loss function Figure 1 The weighting of the consistency loss adopts a cosine annealing weighting strategy (small in the early stage of training and large in the later stage) to avoid the early gradient being dominated by the physical term and affecting convergence.

[0072] Regarding the parameter inversion model of this scheme: A combination of ResNet50 and LSTM is chosen as the basic architecture. ResNet50 captures key local features, LSTM processes temporal information, and a fusion module comprehensively covers various key information in the design, allowing the network to have a more thorough understanding of complex mapping relationships. Based on this, a PINN physical head is connected in parallel: taking the backbone prediction parameters as input, the differentiable "physical head" calculates intermediate physical quantities and applies consistency constraints with prior rules or obtainable approximate targets to obtain the physical loss.

[0073] During training, the parameter inversion model of this scheme feeds a prepared training set into the network, sets an appropriate batch size, learning rate and optimizer, and uses strategies such as learning rate scheduling and early stopping for iteration. The total loss is defined, and each epoch uses a validation / test set to evaluate and record metrics such as accuracy / loss to guide the update of superparticipant weights. In end-to-end backpropagation, the backbone and physical head are updated simultaneously. The physical branches form differentiable prior constraints on the backbone prediction, so that the data-driven results maintain discriminativeness while satisfying physical consistency, thereby improving generalization and robustness.

[0074] This approach tests the parameter inversion model and other model frameworks under the same conditions using a test set. The test results are shown in Table 1 below. Table 1 Comparison of the effects of different network architectures

[0075] As shown in Table 1, the parameter inversion model of this invention has a significantly lower mean squared error (MSE) than the state-of-the-art method (only 0.01), and the number of epochs required for convergence (201) is much less than that of similar methods (which typically require tens of thousands of epochs), demonstrating the dual advantages of this method in terms of accuracy and efficiency.

[0076] This approach trains the parameter inversion model using training datasets of different sizes and tests it using a test set. The performance of the parameter inversion model under different training dataset sizes is shown in the following two figures: Table 2. Performance of parameter inversion models under training datasets of different sizes .

[0077] The network is trained using the constructed training dataset. The trained network can be used to design quasi-bic metamaterial sensors. Subsequently, sensor design-related parameters can be input into the network, and the network will output the corresponding design results.

[0078] Example 3 This solution provides an application method for the parameter inversion model described in Embodiment 2, which is particularly applicable to the design of hollow cylindrical metamaterial sensors.

[0079] Specifically, the application method of this parameter inversion model, namely the parameter design method for hollow cylindrical metamaterial sensors, includes the following steps: The transmission curve of the hollow cylindrical metamaterial sensor is input into the parameter inversion model to obtain the prediction results, which include the asymmetric parameters of the hollow cylindrical metamaterial sensor, the height of the hollow cylindrical silicon structure, the outer ring radius of the hollow cylindrical silicon structure, and the inner ring radius of the hollow cylindrical silicon structure.

[0080] In some embodiments, 1001 sampling points are uniformly taken from the transmission curve to form a one-dimensional sequence of parameters of length 1001, which are then input into the parameter inversion model.

[0081] Example 5 This design proposes a hollow cylindrical metamaterial sensor for hepatitis B virus detection. This sensor exhibits a sharp resonance peak at 0.917 THz. Specifically, the transmission curve of the sharp resonance peak at 0.917 THz is input into the parameter inversion model designed in Example 2 to output the prediction results. The corresponding structural parameters of the hollow cylindrical metamaterial sensor designed in this scheme are shown in Table 3 below. Table 3 shows the structural parameters of the hollow cylindrical metamaterial sensor for hepatitis B virus detection. .

[0082] To accurately obtain the spectral response characteristics of this hollow cylindrical silicon metamaterial sensor, this scheme employs the finite-difference time-domain method in CST software for electromagnetic simulation. In the simulation settings, periodic boundary conditions are used in the x and y directions to reproduce the infinite periodic array characteristics of the structure, while a fully matched layer open boundary is used in the z direction to simulate the free-space radiation environment. All simulations are conducted at room temperature (298.15 K), and the excitation source is a transversely magnetically polarized wave incident perpendicularly from the Zmax port to ensure that the electromagnetic response of the structure under specific polarization can be stably characterized.

[0083] To reveal the excitation mechanism of continuous domain bound states (BIC), this scheme sets the asymmetry parameter to 0, constructs a fully symmetric structure as a control, and compares its transmission spectrum with that of an asymmetric structure with a = 7 μm, as shown in Figure 4. When a = 7 μm, the structure exhibits a sharp Fano resonance peak at 0.917 THz, directly proving that the symmetry-protected BIC can degenerate into an observable quasi-BIC (QBIC) mode under geometric asymmetric perturbation. This resonant frequency falls precisely in the absorption band of the hepatitis B virus characteristic fingerprint, giving this sensor the potential for specific virus detection applications. It also verifies that controllable symmetry breaking can achieve precise control from BIC to QBIC, providing theoretical support for high-performance sensor design.

[0084] in addition, Figure 5 The electromagnetic multipole radiation power distribution under the asymmetric parameter a=7 μm and resonant frequency of 0.917 THz is presented, where TD, MD, MQ, ED, and EQ represent different electromagnetic multipoles. The results show that electric dipoles and magnetic dipoles dominate the contribution, and the resonant mode is a typical electro-magnetic hybrid QBIC mode. This conclusion is highly consistent with the electromagnetic field distribution shown in Figure 6: the electric field is highly localized near the semi-circular gap and the inner and outer walls of the hollow cylinder, forming a significant field enhancement; the magnetic field surrounds the cylinder to form a closed loop, corresponding to a strong displacement current loop and magnetic dipole resonance. The multipole decomposition results reveal the physical nature of the mode and provide a theoretical basis for achieving a high Q value (Q≈3397) and strong field localization in the structure: electric dipoles and magnetic dipoles synergistically enhance the light-matter interaction, and the weak contribution of higher-order multipoles can suppress radiation loss, jointly achieving excellent sensing performance.

[0085] To further elucidate the physical nature of the Fano resonance at 0.917 THz, this scheme analyzes the electromagnetic field distribution at the resonance point. For example... Figure 6As shown in the left figure, electromagnetic energy is strongly localized in specific regions of the central columnar structure, especially in the gap between the two semicircular structures and near the inner and outer walls of the hollow cylinder, exhibiting a significant enhancement effect. This strong localization of the electric field is a hallmark of the excitation of typical electric dipole resonances or dipole-like modes. When the electric field component of the incident electromagnetic wave couples with the asymmetry of the structure, it can drive charge redistribution in these key regions, forming a strong oscillating electric dipole moment. This localized electric field greatly enhances the interaction between light and matter, laying a crucial physical foundation for sensors to detect changes in the refractive index of the medium. The introduction of analytes, such as biomolecules, directly perturbs this locally enhanced electric field, thereby inducing a sensitive shift in the resonant frequency. Figure 6 As shown in the right figure, an energy distribution characteristic complementary to the electric field distribution can be observed. The magnetic field energy mainly surrounds the columnar structure and its hollow region, forming a clear ring-shaped magnetic field line distribution pattern. This characteristic indicates that a strong displacement current loop is induced inside the structure in the resonant state. This loop effectively excites magnetic dipole resonance through its interaction with the incident magnetic field.

[0086] Based on the comprehensive electromagnetic field distribution, the sharp Fano resonance observed at 0.917 THz originates from the structure simultaneously supporting and generating strongly coupled electric and magnetic dipole modes under symmetry-breaking conditions. In an ideal symmetric structure, this mode degenerates into a bound state uncoupled from free-space radiation due to symmetry protection conditions, and cannot be detected in the far field. However, when asymmetry is introduced through the asymmetry parameter 'a', the symmetry is broken, and this bound state mode leaks into an observable QBIC mode, which manifests as a sharp Fano resonance peak in the far-field transmission spectrum. Therefore, Figure 6 The field distribution results not only confirmed the near-field morphology of the QBIC mode, but also revealed the intrinsic physical mechanism by which it generates high-quality factor resonance and ultrasensitive response to the refractive index of the environment, providing a clear physical basis for subsequent sensor parameter inversion based on deep learning.

[0087] To systematically verify the modulatory effect of symmetry breaking on QBIC, this scheme performs parameter scans for different asymmetric parameters 'a', and the results are shown in Figure 7. Figure 8 As shown in the figure. This analysis not only intuitively demonstrates the transformation process from BIC to QBIC, but also reveals the physical laws governing the precise control of the Q value of the resonant mode through geometric parameters.

[0088] Figure 7 and Figure 8The complete physical process of the transformation from symmetry-protected BIC to QBIC is clearly revealed. When the asymmetry parameter a = 0, the metamaterial unit structure is in a perfectly symmetric state, and no resonance peaks are observed in the transmission spectrum near 0.917 THz. This phenomenon corresponds to an ideal symmetry-protected BIC: because its modal distribution is completely decoupled from free-space radiation waves, it theoretically has an infinitely large Q value and cannot be directly detected in the far field, exhibiting a "dark mode" in the spectrum. By introducing the asymmetry parameter a, we effectively break the in-plane symmetry of the structure. This symmetry breaking provides a radiation channel for the BIC mode, causing it to degenerate from the ideal "dark" state into an observable QBIC mode. Figure 8 As shown, a sharp Fano resonance peak appears in the transmission spectrum, which is direct evidence that the symmetry-protected BIC is activated under symmetry breaking. Its typical asymmetric line shape originates from the destructive interference between the discrete QBIC mode and the continuous radiation mode.

[0089] Figure 7 further quantitatively demonstrates that the asymmetry parameter 'a' directly determines the resonant linewidth and Q value: it can be observed that as the asymmetry parameter 'a' increases, the free-wavelength (FWHM) of the resonant peak broadens significantly. According to the definition of Q value, an increase in linewidth directly leads to a decrease in Q value. This phenomenon follows the radiation loss theory of QBIC: the parameter 'a' directly determines the coupling strength between the mode and free space. The larger the value of 'a', the more severe the symmetry breaking, and the stronger the coupling between the mode and the external field, thus increasing radiation loss, manifested as a broadened resonant peak and a decreased Q value. Our study determined that when 'a' = 7 μm, the mode can achieve a maximum Q value of approximately 3397, achieving a relatively optimal balance between high-Q resonant characteristics and observable signal strength.

[0090] In summary, Figure 7 and Figure 8 The theoretical analysis perfectly confirms the physical mechanism of exciting and modulating QBIC through controllable geometric symmetry breaking. This study not only intuitively demonstrates the transformation path from "invisible" BIC to "visible" QBIC, but more importantly, it elucidates an effective way to precisely control the resonant Q value through a simple structural parameter 'a'. This provides crucial theoretical basis and practical design freedom for the subsequent customized design of metamaterial sensors for different application scenarios.

[0091] Based on the confirmed symmetry breaking control mechanism, this scheme further investigates the influence of unbroken geometric parameters on resonance characteristics and evaluates the refractive index sensing performance. Figure 9 and Figure 10 The study reveals the regulation of the resonant frequency by the outer ring radius r1 and the structural height h2, respectively. Although neither of them participates in the symmetry breaking, they can achieve precise frequency tuning of the resonant peak.

[0092] like Figure 9As shown, the resonance peak exhibits a significant redshift as r1 increases. The physical mechanism lies in the fact that increasing r1 directly expands the physical size of the resonant cavity, which, according to the fundamental principles of electromagnetic resonance, leads to a decrease in the resonant frequency. Simultaneously, the increased size is equivalent to enhanced light-matter interaction, increasing the effective refractive index of the mode and further shifting the frequency to lower frequencies. Observations also reveal that the resonance peak linewidth narrows slightly with increasing r1, indicating a slight increase in the Q value. This may be due to a slight decrease in radiation loss resulting from optimized field distribution.

[0093] like Figure 10 As shown, the resonance peak also exhibits a redshift trend as h2 increases. This is because the increased height of the silicon resonator enhances the localization ability of the medium to the light field, thereby significantly increasing the effective refractive index of the mode and ultimately leading to a decrease in the resonant frequency. Notably, the depth and line shape of the resonance peak also vary with h2. The largest transmittance changes were observed at h2 of 42 µm and 46 µm, indicating the existence of an optimal structural height that maximizes the excitation efficiency and coupling strength of the QBIC mode.

[0094] It is evident that by adjusting r1 and h2, this scheme can precisely position the QBIC resonant peak to the target operating band, much like operating a "tuning knob." This provides a solid theoretical and experimental basis for customizing the operating frequency of sensors for specific applications.

[0095] In terms of sensing performance, Figure 11 and Figure 12 The transmission spectrum variation within the environmental refractive index range of 1.0–1.1 is presented. With increasing refractive index, the resonance peak exhibits a regular redshift, stemming from the change in the effective refractive index of the mode as the environmental dielectric constant increases. Calculations show that when the refractive index change Δn = 0.02, the frequency shift Δf = 0.002 THz, corresponding to a sensitivity S = 100 GHz / RIU. To comprehensively evaluate the sensing performance, the quality factor (FOM) is calculated. Under the condition of refractive index n = 1.0, the FOM obtained based on FWHM is as high as 361. This excellent performance is attributed to the synergistic effect of the high Q value and high sensitivity of the QBIC mode, indicating that this sensor simultaneously possesses high response intensity and high resolution, enabling accurate detection of minute refractive index changes.

[0096] To comprehensively evaluate performance, this approach also calculated the FOM (Formula for Memory of Memory) value, which reflects both the sensor's response to changes in refractive index and its ability to distinguish minute variations. Using the FWHM (Frequency of Memory) at a refractive index of 1.0, the sensor achieved a FOM value as high as 361. This excellent FOM value is attributed to the synergistic effect of the inherently high Q value and considerable sensitivity of the QBIC mode, demonstrating that the sensor not only possesses high sensitivity but also excellent resolution.

[0097] Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Anyone skilled in the art can make various modifications and alterations without departing from the spirit and scope of the present invention. Therefore, the scope of protection of the present invention should be determined by the claims.

Claims

1. A method for constructing a parameter inversion model for a hollow cylindrical metamaterial sensor, characterized in that, The design of hollow cylindrical metamaterial sensors includes the following steps: The hollow cylindrical metamaterial sensor includes: at least one metamaterial sensing unit, wherein each metamaterial sensing unit includes a silicon dioxide substrate and at least one hollow cylindrical silicon structure located on the silicon dioxide substrate; each hollow cylindrical silicon structure is a hollow cylindrical ring composed of a first semi-circular ring column and a second semi-circular ring column arranged at intervals, and the second semi-circular ring column is further divided by gaps into symmetrically distributed first arc column and second arc column, the central angle of the first semi-circular ring column is 180°, the central angle of the first arc column and the second arc column are both 90°, and the distance between the first arc column and the second arc column is defined as an asymmetric parameter; S1: Obtain the training dataset, which includes at least one set of transmission curves of hollow cylindrical metamaterial sensors and corresponding hollow cylindrical metamaterial sensor variables for each transmission curve. The hollow cylindrical metamaterial sensor variables include asymmetric parameters, the height of the hollow cylindrical silicon structure, the outer ring radius of the hollow cylindrical silicon structure, and the inner ring radius of the hollow cylindrical silicon structure. S2: The training dataset is input into the parameter inversion framework for training until the loss function meets the set conditions to obtain the parameter inversion model. The parameter inversion framework includes an input layer, a ResNet50 branch, an LSTM branch, a feature fusion module, a linear layer, a PINN physical head, and an output layer. The input layer normalizes the training dataset to obtain one-dimensional sequence data. The one-dimensional sequence data is input into the ResNet50 branch and the LSTM branch to obtain ResNet50 extracted features and LSTM extracted features, respectively. The ResNet50 extracted features and LSTM extracted features are input into the fusion module to output fused features. The fused features are input into the linear layer to obtain the prediction results. The training dataset is input into the PINN physical head to construct the target peak map, and the prediction results are input into the PINN physical head to construct the prediction peak map. The fusion module includes parallel channel and spatial branches. The channel branch comprises global average pooling and global max pooling along the channel dimension, two one-dimensional convolutional layers, and a Softmax function. Features extracted by ResNet50 and LSTM are input into the channel branch, processed by global average pooling and global max pooling along the channel dimension, and then concatenated to obtain aggregated channel features. These aggregated features are then input into two one-dimensional convolutional layers to generate ResNet50 and LSTM channel weights corresponding to the features extracted by ResNet50 and LSTM, respectively. The Softmax function is then used to normalize the ResNet50 and LSTM channel weights, resulting in normalized ResNet50 and LSTM channel weights. The spatial branch comprises global average pooling and global max pooling along the spatial dimension, two one-dimensional convolutional layers, and a Softmax function. Features extracted by ResNet50 and LSTM are input into the spatial branch. The spatial features are obtained by concatenating the global average pooling and global max pooling in the spatial branch. These aggregated spatial features are then input into two 2D convolutional layers to generate ResNet50 spatial weights and LSTM spatial weights corresponding to the features extracted by ResNet50 and LSTM, respectively. The ResNet50 and LSTM spatial weights are then normalized using the Softmax function to obtain normalized ResNet50 and LSTM spatial weights. The normalized ResNet50 channel weights and normalized ResNet50 spatial weights are then multiplied element-wise to obtain the ResNet50 temporal weights. Similarly, the normalized LSTM spatial weights and normalized LSTM channel weights are multiplied element-wise to obtain the LSTM temporal weights. Finally, the ResNet50 and LSTM features are weighted and fused to obtain the fused features. The loss function of the parameter inversion framework is a weighted function of supervised regression loss and peak consistency loss. The supervised regression loss is constructed based on the prediction results and the training dataset, while the peak consistency loss is constructed based on the target peak and the predicted peak.

2. The method for constructing a parameter inversion model for a hollow cylindrical metamaterial sensor according to claim 1, characterized in that, The outer ring radius of the hollow cylindrical silicon structure is 60~80μm, the height of the hollow cylindrical silicon structure is 34~50μm, the inner ring radius of the hollow cylindrical silicon structure is 30~50µm, and the asymmetry parameter is 0~28µm.

3. The method for constructing a parameter inversion model for a hollow cylindrical metamaterial sensor according to claim 1, characterized in that, When the asymmetry parameter is 7 μm, a sharp resonance peak is observed at 0.917 THz when the height of the hollow cylindrical silicon structure is 42 μm, the outer ring radius of the hollow cylindrical silicon structure is 70 μm, and the inner ring radius of the hollow cylindrical silicon structure is 42 μm.

4. The method for constructing a parameter inversion model for a hollow cylindrical metamaterial sensor according to claim 1, characterized in that, The target peak map is obtained by constructing Gaussian kernels and stacking them on multiple local maxima in the transmission spectrum of the training dataset. The calculation formula is shown below: ； in For the target peak diagram, The weight of the k-th peak is calculated based on local peak intensity and sharpness. For the first k The center frequency of a local resonance peak The standard deviation of the Gaussian kernel controls the peak width.

5. The method for constructing a parameter inversion model for a hollow cylindrical metamaterial sensor according to claim 1, characterized in that, The prediction results are input into the lightweight differentiable peak parameter regressor in the PINN physical head to obtain the center frequency and weight of the predicted peak. Then, a predicted peak diagram is constructed based on the center frequency and weight of the predicted peak. The calculation formula is as follows: ； in To predict the peak plot, As weight, To predict the center frequency of the peak, denoted as the standard deviation of the Gaussian kernel.

6. The method for constructing a parameter inversion model for a hollow cylindrical metamaterial sensor according to claim 1, characterized in that, The loss function is a weighted function of the supervised regression loss and the peak plot consistency loss, and the calculation formula is as follows: ； ； ； in To predict the peak plot, For the target peak diagram, For the predicted results, For the true values ​​of the training dataset, For weighted weights, For peak consistency loss, To monitor the regression loss, This is the loss function.

7. The method for constructing a parameter inversion model for a hollow cylindrical metamaterial sensor according to claim 1, characterized in that, The training dataset includes 1001 sampling points uniformly sampled from the transmission curves of at least one set of hollow cylindrical metamaterial sensors, as well as hollow cylindrical metamaterial sensor variables corresponding to each transmission curve.

8. A parameter design method for a hollow cylindrical metamaterial sensor, characterized in that, Includes the following steps: The transmission curve of the hollow cylindrical metamaterial sensor is input into the parameter inversion model to obtain the prediction results, wherein the prediction results are the asymmetric parameters of the hollow cylindrical metamaterial sensor, the height of the hollow cylindrical silicon structure, the outer ring radius of the hollow cylindrical silicon structure, and the inner ring radius of the hollow cylindrical silicon structure. The parameter inversion model is constructed according to the parameter inversion model construction method for hollow cylindrical metamaterial sensors according to any one of claims 1 to 7.

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